Those clamps you mentioned output voltage, so you should connect those first to C and V inputs, at least for now.
If your max currents would be under 1000 (1 mV/A = 1 volt out) or 100 (10 mV/A = 1 volt out) amps, so outputs would stay at under 1 volt, you might want to try auxiliary inputs, which gives you somewhat better resolution with those millivolts. In that case, you would connect you clamp to C and ohm inputs and select aux voltage from mooshimeter interface.
Sadly, there are no way to set mooshimeter to do your math, so you just need to mentally move your decimal point to right place and remember that those mV:s represent amperes.
A-input has nothing to do with the measurement above, however, you could use it as another common input, as A and C only have big fuse and really small resistance between them. So you might have line voltage between V and A inputs (which feels wrong, but then those common leads would not cross each other) and your millivolt output current clamp between C and ohm. Just be very sure that you know what you are doing, it is very easy to fry your multimeter with that many leads connected to it! It is very easy to even get your current clamp energized to line voltage, if you are not sure which lead is live and which is neutral. Take your time.
Then there are current clamps that output (milli-) amperes, but those are for AC only. Those would go to C and A inputs and you likely get a bit lower resolution out from your multimeter with those. I have one with 1000:1 ratio, so it is easy to read, just skip the milli-part from results. Works great with tens of amperes, under one amp it disappears to noise.